Mathematics | Senioritis

John Rhodes
June 21, 2023
Linear Algebra

Infinite Inner Products On R^N: Debunking The Common Misconception – A Comprehensive Guide On Varying Definitions Of Inner Products In Vector Spaces

there is exactly one inner product on the vector space R^n 0 This statement is not true. In fact, there are infinitely many inner products on the...

John Rhodes
June 21, 2023
Linear Algebra

Inner Products: Linearity, Conjugate Symmetry, And Positive Definiteness In Vector Spaces

an inner product is a scalar valued function on the set of ordered pairs 1 of vectors in a vector space that satisfies three key properties: linearity...

John Rhodes
June 21, 2023
Linear Algebra

The Limitations Of The Dot Product: When = Then Y=Z Is Not True

if = then y=z in general 0 The statement = then y=z is not true in general. In general, = x1y1 + x2y2 + … + xnyn,...

John Rhodes
June 21, 2023
Linear Algebra

Addition Of Elementary Matrices: Properties And Examples In Linear Algebra

the sum of two n x n elementary matrices is still an elementary matrix 0 Let’s first define what an elementary matrix is. An n x n...

John Rhodes
June 21, 2023
Linear Algebra

Exploring Non-Elementary Matrices: Why The Product Of Two Elementary Matrices Is Not Always Elementary

the product of two n x n elementary matrices in general is not necessarily an elementary matrix 1 An elementary matrix is a matrix obtained from an...

John Rhodes
June 21, 2023
Linear Algebra

Simplifying Determinants Of Matrices: How To Calculate Determinants Of Transpose And Higher Powers Of A Matrix

for any n x n matrix A detA^tdetA 0 We can begin by using the properties of the determinant to simplify the given expression: det(A^T) = det(A)...

John Rhodes
June 21, 2023
Linear Algebra

Elementary Column Operations: How They Simplify And Transform Matrices Without Affecting Determinant And Eigenvalues

elementary column operations do not necessarily preserve rank 0 Elementary column operations are operations that we perform on the columns of a matrix to simplify or transform...

John Rhodes
June 21, 2023
Linear Algebra

Matrix Addition For Linear Transformations: Proof That [T+U] = [T]+[U] With Given Ordered Bases

let V and W be finite dimensional vector spaces with ordered bases beta and lamda respectively. let T, U: V –> W be linear transformations then [t+U]...